Free space (electromagnetism): Difference between revisions

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imported>John R. Brews
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imported>John R. Brews
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{{cite book |title=Fundamentals of quantum optics and quantum information |author=Peter Lambropoulos, David Petrosyan |url=http://books.google.com/books?id=53bpU-41U8gC&pg=PA30 |pages=p. 30 |isbn=354034571X |year=2007 |publisher=Springer}}
{{cite book |title=Fundamentals of quantum optics and quantum information |author=Peter Lambropoulos, David Petrosyan |url=http://books.google.com/books?id=53bpU-41U8gC&pg=PA30 |pages=p. 30 |isbn=354034571X |year=2007 |publisher=Springer}}


</ref> In particular, it applies also to the electromagnetic field. A digression follows to flesh out the role of commutators for the electromagnetic field.<ref name=Vogel>
</ref> In particular, it applies also to the electromagnetic field. A digression follows to flesh out the role of commutators for the electromagnetic field.<ref name=Vogel2>


{{cite book |title=Quantum optics |author=Werner Vogel, Dirk-Gunnar Welsch |pages=pp. 18 ''ff'' |chapter= Chapter 2: Elements of quantum electrodynamics |isbn=3527405070 |year=2006 |publisher=Wiley-VCH |edition =3rd ed. |url=http://books.google.com/books?id=eouwERvRrJEC&pg=PA18&lpg=PA18 }}
{{cite book |title=Quantum optics |author=Werner Vogel, Dirk-Gunnar Welsch |pages=pp. 18 ''ff'' |chapter= Chapter 2: Elements of quantum electrodynamics |isbn=3527405070 |year=2006 |publisher=Wiley-VCH |edition =3rd ed. |url=http://books.google.com/books?id=eouwERvRrJEC&pg=PA18&lpg=PA18 }}
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:The standard approach to the quantization of the electromagnetic field begins by introducing a ''vector'' potential '''A''' and a scalar potential ''V'' to represent the basic electromagnetic electric field '''E''' and magnetic field '''B''' using the relations:<ref name=Vogel/>
:The standard approach to the quantization of the electromagnetic field begins by introducing a ''vector'' potential '''A''' and a scalar potential ''V'' to represent the basic electromagnetic electric field '''E''' and magnetic field '''B''' using the relations:<ref name=Vogel2/>


:::<math>\mathbf B =\mathbf {\nabla  \times A} \ </math>
:::<math>\mathbf B =\mathbf {\nabla  \times A} \ </math>

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Free space usually refers to a perfect vacuum, devoid of all particles. The term is most often used in classical electromagnetism where it refers to a reference state,[1] and in quantum physics where it refers to the ground state of the electromagnetic field, which is subject to fluctuations about a dormant zero average-field condition.[2] The classical case of vanishing fields implies all fields are source-attributed, while in the quantum case field moments can arise without sources by virtual phonon creation and destruction.[3] The description of free space varies somewhat among authors, with some authors requiring only the absence of substances with electrical properties,[4] or of charged matter (ions and electrons, for example).[5]

Classical case

In classical physics, free space is a concept of electromagnetic theory, corresponding to a theoretically perfect vacuum and sometimes referred to as the vacuum of free space, or as classical vacuum, and is appropriately viewed as a reference medium.[1] In the classical case, free space is characterized by the electrical permittivity ε0 and the magnetic permeability μ0.[6] The exact value of ε0 is provided by NIST as the electric constant and the defined value of μ0 as the magnetic constant:

ε0 ≈ 8.854 187 817... × 10−12 F m−1
μ0 = 4π × 10−7 ≈ 12.566 370 614... x 10−7 N A−2

where the approximation is not a physical uncertainty (such as a measurement error) but a result of the inability to express these irrational numbers with a finite number of digits. The SI units farad, metre, newton and ampere are denoted by 'F', 'm', 'N', and 'A'.

One consequence of these electromagnetic properties coupled with Maxwell's equations is that the speed of light in free space is related to ε0 and μ0 via the relation:[7]

Using the defined valued for the speed of light provided by NIST as:

c0 = 299 792 458 m s −1,

and the already mentioned defined value for μ0, this relationship leads to the exact value given above for ε0.

Another consequence of these electromagnetic properties is that the ratio of electric to magnetic field strengths in an electromagnetic wave propagating in free space is an exact value provided by NIST as the characteristic impedance of free space:

= 376.730 313 461... ohms.

It also can be noted that the electrical permittivity ε0 and the magnetic permeability μ0 do not depend upon direction, field strength, polarization, or frequency. Consequently, free space is isotropic, linear, non-dichroic, and dispersion free. Linearity, in particular, implies that the fields and/or potentials due to an assembly of charges is simply the addition of the fields/potentials due to each charge separately (that is, the principle of superposition applies).[8]

Quantum case

The Heisenberg uncertainty principle for a particle in one dimension does not allow a state of rest in which the particle is simultaneously at a fixed location, say the origin of coordinates, and has also zero momentum. Instead the particle has a zero-point energy and a range of momentum and spread in location attributable to quantum fluctuations.

An uncertainty principle applies to all quantum mechanical operators that do not commute.[9] In particular, it applies also to the electromagnetic field. A digression follows to flesh out the role of commutators for the electromagnetic field.[10]

The standard approach to the quantization of the electromagnetic field begins by introducing a vector potential A and a scalar potential V to represent the basic electromagnetic electric field E and magnetic field B using the relations:[10]
The vector potential is not completely determined by these relations, leaving open a so-called gauge freedom. Resolving this ambiguity using the Coulomb gauge leads to a description of the electromagnetic fields in the absence of charges in terms of the vector potential and the momentum field Π, given by:
Quantization is achieved by insisting that the momentum field and the vector potential do not commute.

Because of the non-commutation of field variables, the variances of the fields cannot be zero, although their averages are zero.[11] As a result, the vacuum can be considered as a dielectric medium, and is capable of vacuum polarization.[12] Its electrical permittivity can be calculated, and it differs slightly from the simple ε0 of the classical vacuum. Likewise, its permeability can be calculated and differs slightly from μ0.

Attainability

A perfect vacuum is itself only realizable in principle.[13][14] It is an idealization, like absolute zero for temperature, that can be approached, but never actually realized. And classical vacuum is one step further removed from attainability because its permittivity ε0 and permeability μ0 do not allow for quantum fluctuation effects. Nonetheless, outer space and good terrestrial vacuums are modeled adequately by classical vacuum for many purposes.

References

  1. ↑ 1.0 1.1 Werner S. Weiglhofer and Akhlesh Lakhtakia (2003). “§4.1: The classical vacuum as reference medium”, Introduction to complex mediums for optics and electromagnetics. SPIE Press. ISBN 0819449474. 
  2. ↑ Ramamurti Shankar (1994). Principles of quantum mechanics, 2nd ed.. Springer, p. 507. ISBN 0306447908. 
  3. ↑ Werner Vogel, Dirk-Gunnar Welsch (2006). Quantum optics, 3rd ed.. Wiley-VCH, p. 337. ISBN 3527405070. 
  4. ↑ RK Pathria (2003). The Theory of Relativity, Reprint of Hindustan 1974 2nd ed.. Courier Dover Publications, p. 119. ISBN 0486428192. 
  5. ↑ (1992) Christopher G. Morris, editor: Academic Press dictionary of science and technology. Academic, p. 880. ISBN 0122004000. 
  6. ↑ Akhlesh Lakhtakia, R. Messier (2005). “§6.2: Constitutive relations”, Sculptured thin films: nanoengineered morphology and optics. SPIE Press, p. 105. ISBN 0819456063. 
  7. ↑ Albrecht Unsöld, B. Baschek (2001). “§4.1: Electromagnetic radiation, Equation 4.3”, The new cosmos: an introduction to astronomy and astrophysics, 5th ed.. Springer, p. 101. ISBN 3540678778. 
  8. ↑ A. Pramanik (2004). “§1.3 The principle of superposition”, Electro-Magnetism: Theory and Applications. PHI Learning Pvt. Ltd, pp. 37-38. ISBN 8120319575. 
  9. ↑ Peter Lambropoulos, David Petrosyan (2007). Fundamentals of quantum optics and quantum information. Springer, p. 30. ISBN 354034571X. 
  10. ↑ 10.0 10.1 Werner Vogel, Dirk-Gunnar Welsch (2006). “Chapter 2: Elements of quantum electrodynamics”, Quantum optics, 3rd ed.. Wiley-VCH, pp. 18 ff. ISBN 3527405070. 
  11. ↑ Gilbert Grynberg, Alain Aspect, Claude Fabre (2010). Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light. Cambridge University Press, pp. 351 ff. ISBN 0521551129. 
  12. ↑ Kurt Gottfried, Victor Frederick Weisskopf (1986). Concepts of particle physics, Volume 2. Oxford University Press, 259 ff. ISBN 0195033930. 
  13. ↑ Luciano Boi (2009). Ernesto Carafoli, Gian Antonio Danieli, Giuseppe O. Longo, editors: The Two Cultures: Shared Problems. Springer, p. 55. ISBN 8847008689. 
  14. ↑ PAM Dirac (2001). Jong-Ping Hsu, Yuanzhong Zhang, editors: Lorentz and Poincaré invariance: 100 years of relativity. World Scientific, p. 440. ISBN 9810247214. 
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